Critical exponent for the semilinear wave equations with a damping increasing in the far field

Abstract

We consider the Cauchy problem of the semilinear wave equation with a damping term \begin{align*} u_{tt} - Δu + c(t,x) u_t = |u|^p, \quad (t,x)\in (0,\infty)\times \mathbb{R}^N,\quad u(0,x) = \varepsilon u_0(x), \ u_t(0,x) = \varepsilon u_1(x), \quad x\in \mathbb{R}^N, \end{align*} where $p>1$ and the coefficient of the damping term has the form \begin{align*} c(t,x) = a_0 (1+|x|^2)^{-α/2} (1+t)^{-β} \end{align*} with some $a_0 > 0$, $α< 0$, $β\in (-1, 1]$. In particular, we mainly consider the cases $ α< 0, β=0$ or $α< 0, β= 1$, which imply $α+ β< 1$, namely, the damping is spatially increasing and effective. Our aim is to prove that the critical exponent is given by $ p = 1+ \frac{2}{N-α}$. This shows that the critical exponent is the same as that of the corresponding parabolic equation $c(t,x) v_t - Δv = |v|^p$. The global existence part is proved by a weighted energy estimates with an exponential-type weight function and a special case of the Caffarelli-Kohn-Nirenberg inequality. The blow-up part is proved by a test-function method introduced by Ikeda and Sobajima (arXiv:1710.06780v1). We also give an upper estimate of the lifespan.